wlkiswwlksupgr2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > wlkiswwlksupgr2		Structured version Visualization version GIF version

Theorem wlkiswwlksupgr2 26763

Description: A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 26761 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 9302) for its proof. Notice that only the existence of a function 𝑓 can be proven, but, in general, it cannot be "constructed" (as in wlkiswwlks2 26761). (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)

**Assertion**
Ref	Expression
wlkiswwlksupgr2	⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))

Distinct variable groups: 𝑓,𝐺 𝑃,𝑓

Proof of Theorem wlkiswwlksupgr2 Dummy variables 𝑖 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2		eqid 2622	. . 3 ⊢ (Edg‘𝐺) = (Edg‘𝐺)
3	1, 2	iswwlks 26728	. 2 ⊢ (𝑃 ∈ (WWalks‘𝐺) ↔ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)))
4		edgval 25941	. . . . . . . . . . . . 13 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺)
5	4	eleq2i 2693	. . . . . . . . . . . 12 ⊢ ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺))
6		upgruhgr 25997	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
7		eqid 2622	. . . . . . . . . . . . . . . 16 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	7	uhgrfun 25961	. . . . . . . . . . . . . . 15 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
9	6, 8	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UPGraph → Fun (iEdg‘𝐺))
10	9	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) → Fun (iEdg‘𝐺))
11		elrnrexdm 6363	. . . . . . . . . . . . . 14 ⊢ (Fun (iEdg‘𝐺) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥)))
12		eqcom 2629	. . . . . . . . . . . . . . 15 ⊢ (((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥))
13	12	rexbii 3041	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∃𝑥 ∈ dom (iEdg‘𝐺){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} = ((iEdg‘𝐺)‘𝑥))
14	11, 13	syl6ibr 242	. . . . . . . . . . . . 13 ⊢ (Fun (iEdg‘𝐺) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
15	10, 14	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran (iEdg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
16	5, 15	syl5bi 232	. . . . . . . . . . 11 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) → ({(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
17	16	ralimdv 2963	. . . . . . . . . 10 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝐺 ∈ UPGraph ) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^((#‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
18	17	ex 450	. . . . . . . . 9 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝐺 ∈ UPGraph → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → ∀𝑖 ∈ (0..^((#‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
19	18	com23 86	. . . . . . . 8 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^((#‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
20	19	3impia 1261	. . . . . . 7 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝐺 ∈ UPGraph → ∀𝑖 ∈ (0..^((#‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
21	20	impcom 446	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ∀𝑖 ∈ (0..^((#‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
22		ovex 6678	. . . . . . 7 ⊢ (0..^((#‘𝑃) − 1)) ∈ V
23		fvex 6201	. . . . . . . 8 ⊢ (iEdg‘𝐺) ∈ V
24	23	dmex 7099	. . . . . . 7 ⊢ dom (iEdg‘𝐺) ∈ V
25		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = (𝑓‘𝑖) → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘(𝑓‘𝑖)))
26	25	eqeq1d 2624	. . . . . . 7 ⊢ (𝑥 = (𝑓‘𝑖) → (((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
27	22, 24, 26	ac6 9302	. . . . . 6 ⊢ (∀𝑖 ∈ (0..^((#‘𝑃) − 1))∃𝑥 ∈ dom (iEdg‘𝐺)((iEdg‘𝐺)‘𝑥) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} → ∃𝑓(𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
28	21, 27	syl 17	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ∃𝑓(𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
29		iswrdi 13309	. . . . . . . . . 10 ⊢ (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑓 ∈ Word dom (iEdg‘𝐺))
30	29	adantr 481	. . . . . . . . 9 ⊢ ((𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → 𝑓 ∈ Word dom (iEdg‘𝐺))
31	30	adantl 482	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → 𝑓 ∈ Word dom (iEdg‘𝐺))
32		wrdfin 13323	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → 𝑃 ∈ Fin)
33		hashnncl 13157	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ Fin → ((#‘𝑃) ∈ ℕ ↔ 𝑃 ≠ ∅))
34	33	bicomd 213	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Fin → (𝑃 ≠ ∅ ↔ (#‘𝑃) ∈ ℕ))
35	32, 34	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → (𝑃 ≠ ∅ ↔ (#‘𝑃) ∈ ℕ))
36	35	biimpac 503	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (#‘𝑃) ∈ ℕ)
37		wrdf 13310	. . . . . . . . . . . . . . . 16 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → 𝑃:(0..^(#‘𝑃))⟶(Vtx‘𝐺))
38		nnz 11399	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑃) ∈ ℕ → (#‘𝑃) ∈ ℤ)
39		fzoval 12471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((#‘𝑃) ∈ ℤ → (0..^(#‘𝑃)) = (0...((#‘𝑃) − 1)))
40	38, 39	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑃) ∈ ℕ → (0..^(#‘𝑃)) = (0...((#‘𝑃) − 1)))
41	40	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(#‘𝑃)) = (0...((#‘𝑃) − 1)))
42		nnm1nn0 11334	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((#‘𝑃) ∈ ℕ → ((#‘𝑃) − 1) ∈ ℕ₀)
43		fnfzo0hash 13234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((#‘𝑃) − 1) ∈ ℕ₀ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (#‘𝑓) = ((#‘𝑃) − 1))
44	42, 43	sylan 488	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (#‘𝑓) = ((#‘𝑃) − 1))
45	44	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → ((#‘𝑃) − 1) = (#‘𝑓))
46	45	oveq2d 6666	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0...((#‘𝑃) − 1)) = (0...(#‘𝑓)))
47	41, 46	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(#‘𝑃)) = (0...(#‘𝑓)))
48	47	feq2d 6031	. . . . . . . . . . . . . . . . . 18 ⊢ (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (𝑃:(0..^(#‘𝑃))⟶(Vtx‘𝐺) ↔ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
49	48	biimpcd 239	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃:(0..^(#‘𝑃))⟶(Vtx‘𝐺) → (((#‘𝑃) ∈ ℕ ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
50	49	expd 452	. . . . . . . . . . . . . . . 16 ⊢ (𝑃:(0..^(#‘𝑃))⟶(Vtx‘𝐺) → ((#‘𝑃) ∈ ℕ → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺))))
51	37, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑃 ∈ Word (Vtx‘𝐺) → ((#‘𝑃) ∈ ℕ → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺))))
52	51	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → ((#‘𝑃) ∈ ℕ → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺))))
53	36, 52	mpd 15	. . . . . . . . . . . . 13 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
54	53	3adant3 1081	. . . . . . . . . . . 12 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
55	54	adantl 482	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
56	55	com12 32	. . . . . . . . . 10 ⊢ (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
57	56	adantr 481	. . . . . . . . 9 ⊢ ((𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺)))
58	57	impcom 446	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺))
59		simpr 477	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
60	36, 44	sylan 488	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (#‘𝑓) = ((#‘𝑃) − 1))
61	60	oveq2d 6666	. . . . . . . . . . . . . . . 16 ⊢ (((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(#‘𝑓)) = (0..^((#‘𝑃) − 1)))
62	61	ex 450	. . . . . . . . . . . . . . 15 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺)) → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → (0..^(#‘𝑓)) = (0..^((#‘𝑃) − 1))))
63	62	3adant3 1081	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → (0..^(#‘𝑓)) = (0..^((#‘𝑃) − 1))))
64	63	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) → (0..^(#‘𝑓)) = (0..^((#‘𝑃) − 1))))
65	64	imp 445	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) → (0..^(#‘𝑓)) = (0..^((#‘𝑃) − 1)))
66	65	adantr 481	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (0..^(#‘𝑓)) = (0..^((#‘𝑃) − 1)))
67	66	raleqdv 3144	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ↔ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
68	59, 67	mpbird 247	. . . . . . . . 9 ⊢ ((((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ 𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺)) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
69	68	anasss 679	. . . . . . . 8 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})
70	31, 58, 69	3jca 1242	. . . . . . 7 ⊢ (((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) ∧ (𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})) → (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
71	70	ex 450	. . . . . 6 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ((𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
72	71	eximdv 1846	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (∃𝑓(𝑓:(0..^((#‘𝑃) − 1))⟶dom (iEdg‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
73	28, 72	mpd 15	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
74	1, 7	upgriswlk 26537	. . . . . 6 ⊢ (𝐺 ∈ UPGraph → (𝑓(Walks‘𝐺)𝑃 ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
75	74	adantr 481	. . . . 5 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (𝑓(Walks‘𝐺)𝑃 ↔ (𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
76	75	exbidv 1850	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → (∃𝑓 𝑓(Walks‘𝐺)𝑃 ↔ ∃𝑓(𝑓 ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘𝑓))⟶(Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^(#‘𝑓))((iEdg‘𝐺)‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
77	73, 76	mpbird 247	. . 3 ⊢ ((𝐺 ∈ UPGraph ∧ (𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺))) → ∃𝑓 𝑓(Walks‘𝐺)𝑃)
78	77	ex 450	. 2 ⊢ (𝐺 ∈ UPGraph → ((𝑃 ≠ ∅ ∧ 𝑃 ∈ Word (Vtx‘𝐺) ∧ ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺)) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
79	3, 78	syl5bi 232	1 ⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 ∅c0 3915 {cpr 4179 class class class wbr 4653 dom cdm 5114 ran crn 5115 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 0cc0 9936 1c1 9937 + caddc 9939 − cmin 10266 ℕcn 11020 ℕ₀cn0 11292 ℤcz 11377 ...cfz 12326 ..^cfzo 12465 #chash 13117 Word cword 13291 Vtxcvtx 25874 iEdgciedg 25875 Edgcedg 25939 UHGraph cuhgr 25951 UPGraph cupgr 25975 Walkscwlks 26492 WWalkscwwlks 26717

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-ac2 9285 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ifp 1013 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-ac 8939 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-edg 25940 df-uhgr 25953 df-upgr 25977 df-wlks 26495 df-wwlks 26722

This theorem is referenced by: wlkiswwlkupgr 26764 wlklnwwlklnupgr2 26771

W3C validator